Encoding

  • Write the codeword as x1 x2 x3 x4 … where needed, substitute in the message
  • Multiply H by the codeword, equate each row to 0
  • Solve to find the check-bits
  • Merge

Decoding

  • Multiply H by the message (as a vector)
  • If the resulting vector is
    • Zero vector - no errors, continue
    • Non-zero - Fix the bit associated with the index of the resulting vector in the parity matrix

  • Strip check-bits to extract message

  • **If the resulting vector is a multiple of one of the columns in the parity matrix, substract the related bit n times, such that the resulting vector divides the parity column n times.

Codewords in a ternary generator matrix: 3^n rows

Codewords in the basis of a binary linear code with parity check matrix H. (minimum number of elements needed to create all other) (h without the I matrix?)

Greatest number of information bits for a radix 3 2 error linear code with m=5 check bits.

Given a parity check matrix H, the minimum distance d© of the code is the minimal weight.

Minimal (Hamming) distance = minimum number of changes == smallest number of linearly dependent columns

Read lecture three